How to estimate a genome's size?
Gel electrophoresis is a basic lab method to get size info about nucleotide fragments, although one might need to chop up larger DNA to be able to apply this method. The wikipedia page covers the topic well, thus I'd suggest reading it. I'll just give a short outline:


Nucleotide fragments are loaded into a gel (agarose or acrylamide). 
The gel is put into a static electric field 
Nucleotide fragment move to the positive end, because they have negative charge
larger fragments migrate slower than smaller fragments
standardized DNA ladder can be used to get approx size.


UV fluorescent dye is used to stain the fragments.

Another way, is to check closely related species. Among closely related species it is a fair assumption that they have similar genome size. You can also hybridize the unknown DNA to a DNA of such close relative to see how much of them are similar and you can also make estimates based on the size of the hybridized regions.