Why does my gel have such poor resolution?
Although your tech is referring to your DNA samples, your ladder also indicates there is some room for improvement of your running technique. -20C for a week should not have impacted the quality of the DNA assuming that it was good quality going into the freezer. You can mitigate DNAase activity by adding EDTA to your storage buffer, which will pull metal ions out of solution. Just make sure to take that into account if you're doing a reaction later that requires metal ions (like PCR).

DNA

Focusing first on the DNA samples. There are a few things to check.

How much did you load? I see you stated 20uL, but what was the concentration? It it more important to know the amount of ng being used as the sample. Various stains have different concentrations they can detect (EtBr, SYBR, etc.) I would check the specifics of the stain you're using to make sure you're loading an appropriate amount of DNA.

Are these PCR products? If so, how many cycles was your PCR run? I have seen similar results from PCR if the samples are run for too many cycles. Non-specific products begin to be produced in later cycles, and will cause degradation of your specific products. I would also re-check your primers, the extension time, and the polymerase you're using. It is more difficult to do extensions of the 5.8kb and 6.8kb sizes you mentioned, but not impossible.

Are these restriction digests? Do you have a control well showing the undigested plasmid, or other source? Your source should have a clean band first to expect a clean band in the results. If your source is clean and bright, and the digest looks like the samples above, check your digest buffer and enzymes.

Electrophoresis

There are several things which can improve the quality of an agarose gel. Thermofisher provides a guide for their tips and tricks. Below are things that I have found useful in my own experience.


The time between casting the gel and running it. A common practice is to make gels and store them in TBE/TAE buffer until they're ready to be used. In my experience, the quality of the gel will degrade over time, yielding diffuse bands. The best results are achieved with a freshly poured and cast gel.
Loading technique. Carefully loading the samples into the wells so that they settle as closely to the bottom without mixing with the surrounding buffer is key. The best results are achieved by having your pipet tip touching the bottom of the well, and slowly pulling it out as you release your sample so the sample does not have to settle down into the well on its own.
Run immediately after loading, and image immediately after running. Diffusion will cause nice tight bands to become poor the longer the samples sit in the well.
Voltage, timing, and agarose concentration. Getting these right takes practice. You'll want to try to run your gel as quickly as possible. The more time it sits in the gel, the more diffusion will occur. However, have the voltage too high, and you risk melting your gel. Vary the concentration of agarose to target the size of DNA you're after. I have previously used anywhere between 0.5% and 1.5%.
Prestaining the gel or poststaining? I prefer prestaining for the stains that work well, as it reduces the time the samples will sit in the gel allowing diffusion to occur.
I am looking at your ladder, and the link to the provided standard, and it is difficult to tell which bands are which. The 3.0kb fragment should stand out as the brightest, however the bottom band in your gel is the brightest. Is this the 3.0kb band? Maybe. It's difficult to count the number of bands above it to confirm. If it's not, which might be indicated by the fact that none of your samples seem to have run to the end of the gel, then this would be the 0.5kb band. If it is the 0.5kb band, the ladder intensity doesn't match up with your expected intensity. Double check your ladder so you can know what sizes you're looking at.