Quantifying Gene Expression
What is Protein Expression Level?
This was the original title of the post, which I edited myself because I regard the answer as trivial, but the question as more substantial. To deal with the trivial first:

‘Level’ is not a scientific unit, and can only be used unambiguously
as a scientific term in its English sense in relation to liquids, e.g. “The level of mercury in the thermometer had
fallen.”, “The land is 10 metres above sea level.”
It is used by some people in informal speech or writing to indicate an
undefined quantitation, and because of its very ambiguity is to be
strongly discouraged in scientific communication.

So neither I nor anyone else can say what ‘protein expression level’ is without finding out what the originator of the phrase intended (if indeed he knew) in any individual instance: quantity or concentration of protein, or the rate of its synthesis.
Quantitation
Let us consider some general aspects of a cellular molecule that may require quantitation: Amount, Rate of Synthesis, Rate of Degradation and, where relevant, Biological Activity.
The quantitation of molecules at its most basic is the number species,  or, more practically, their total mass (grams), where possible related to their molecular mass (i.e. moles). The rate of their synthesis or degradation is expressed in terms of the change of their amount in unit time.
In order to compare different systems a reference is needed for the quantitation. This may be per unit volume, per cell, per g cellular protein, per g DNA etc. (However for comparison of similar systems or within one system the reference is often omitted.)
The biological activity of molecules only has meaning if the molecules are, indeed, biologically active (e.g. enzymes). It is expressed in units related to that activity.
Examples of Quantitation for Proteins
The actual units used in quantitation are determined by how one is able to measure the parameter of interest.
Relative amount of protein: You might detect a protein by the intensity of staining of a band on a gel, or by the extent of precipitation using a corresponding antibody. These would have to be calibrated against standards in order to convert raw experimental measurements to g or mole protein. Typical units of relative amount are g/g total protein, g/g DNA.
Rate of synthesis or degradation of protein. You might detect the synthesis of protein by the rate of incorporation of radioactive amino acids into non-radioactive protein, and the rate of degradation by the liberation of radioactive amino acids from pre-labelled protein. Typical units of synthesis would be mg amino acid incorporated per min per mg total protein. (This can be converted to molecules synthesized or degraded per minute, if necessary.)
‘Protein Activity’: The use of this term is not recommended. In chemical terms the activity of a molecule is a measure of its “effective concentration”, and in relation to proteins would only be the concern of biophysicists and the like. Other biologists would associate the term with the biological activity a protein such as an enzyme might have, but as many proteins are structural the term ‘activity’ cannot be applied to proteins generally. Where it is of interest it would be quantified in terms of the nature of the activity, e.g. an enzyme is quantified in units related to the amount of substrate converted to product in a given time.

Quantitation of Gene Expression using Oligonucleotide Microarrays
The two extremes in quantifying gene expression are the detailed study of the expression of one particular gene encoding a protein for which information and tools are available; and general study of the expression of many genes using modern methods that allow the study of many molecules simultaneously. Such methods included mass spectrometry (for small metabolites), two-dimensional gel electrophoresis (for proteins) and microarrays or RNAseq (for mRNA). In general in the latter case one would examine the effect of some agent or condition on the whole spectrum of gene expression.
Let us examine microarray technology as that is the main concern of the poster.

Microarray technology measures the relative amounts of mRNA.

The methodology involves amplifying a mixture mRNA species by reverse transcription to cDNAs, which are labelling with a fluorescent dye. The cDNA is hybridized to immobilized oligonucleotides based on the sequences of the genes in the organism, and the strength of the fluorescence signal is assumed to be proportional to the amount of individual mRNA species in the sample. However the data supplied to the user is the relative image intensity within the experiment, rather than units that allow one to calculate the actual amount of mRNA.
How does these relative amounts of mRNA relate to gene expression?

The relative amounts of mRNA are a relative measure of the rate of synthesis of proteins (3 in dig.), if one assumes that the rate of synthesis of each protein is similarly proportional to the amount of its mRNA (a in dig.) which would be limiting. This is a reasonable assumption in most cases given mRNA is degraded more rapidly than protein. (They are not a measure of the amount of protein — b in dig.)

The relative amounts of mRNA cannot be taken as a measure of the rate of transcription of a mRNA from its gene (1 in dig.) because of the relatively greater effect of mRNA degradation (2 in dig.) on the steady state concentration of mRNAs, and the fact that different mRNAs have different half-lives.


Postscript for the Poster
I have tried to make this answer general, to be of use to more people. As the poster is not a biologist she may still have difficulty in understanding the background to the biological experiments that generated the data she is analysing.
The biological system of interest is gene expression, which encompasses the whole series of events from transcription of genes to mRNA and their translation into protein. The background to this is that there are some genes that are (almost) always expressed, irrespective of physiological circumstances because they are needed to maintain the structure and everyday functions of the cell. This is termed constitutive expression, examples being expression of the genes for cytoskeletal actins or ribosomal proteins. Other genes are expressed (‘switched on’) only when needed (and may be termed inducible). The question here would appear to be whether the genes for the enzymes involved in DNA repair are expressed all the time (constitutively) to deal with normal ‘wear and tear’ of DNA, or whether their expression only occurs in response to some insult knwn to damage DNA, such as gamma irradiation.
As I mentioned, as an experimental scientist, one approach that can be adopted here is to look at the expression of one or two well characterized proteins that are known to be involved in DNA repair. However, there may be proteins involved in this process that you are unaware of, so the modern methods of looking at the expression of all the genes in an organism (if the DNA sequence is known) — oligonucleotide microarrays or, better, RNASeq. These methods measure relative quantities of mRNA in a cell. This is not a proxy for amount of protein or rate of  its synthesis (the term ‘protein activity’ is meaningless and is not and should not be used), it is what it is, but is also a reflection of the expression of the genes that encode the mRNAs. No expression, no mRNA.
You can think of the microarray and RNASeq approaches as a fishing expeditions. If you find mRNAs that are only present after the cell receives a stimulus or an insult, expression of the mRNA has occurred (whether or not the amounts are directly proportional to the rate of synthesis). If the mRNA is synthesised you can assume it is translated into the protein it encodes. In the case of gamma irradiation you might assume any mRNA that shows a large increase in quantity encodes a protein that involved in protecting the cell from the radiation. This will be of scientific interest, especially if it is not what one anticipated.