Until recently I had a real passion for collecting ancient coins, but my finances don't allow me to continue this hobby

Losing enjoyment in activities you previously did enjoy is a very common symptom of a depressive episode (as is everything else you've said), so please contact your doctor and see if you can get assessed. I don't know what country you're in, but I hope you're somewhere you're able to access the help you need.

I'm sure if your leg stopped working you'd go see the doctor, this is no different.

and my family is also quite dismissive of it in general

Who gives a shit what they think. If you enjoy something, and it doesn't harm anyone else, then why should their opinion matter?

Last week I actually dropped a denarius of Tiberius (also known as a "tribute penny") and it shattered. I destroyed a 2000-year old artifact that I loved with all my heart, and since I haven't touched any other coins or looked at any online.

This sucks, but it wasn't your fault. Accidents happen, and you're not the first, or last, person who will have done this. I understand why you feel like shit, and it's OK to feel like shit when something bad happens, but you can also forgive yourself for mistakes.

I used to think I was "good with computers", but almost a decade of working in the field have shown me that no, I'm a low level amateur not even close to some 20-year olds who are professionals in the niche.

You're clearly not an amateur if you have 10 years professional experience. You're literally a professional by definition, your current skill level is valuable, and proves you are indeed good with computers.

Comparison is the thief of joy. No matter what you do, statistically speaking there will always be people better than you, and people worse than you. If you define good by the abilities of the exceptional you will most likely never be good enough. By all means identify areas you can improve, and even use other people as role models for that area, but artificially reducing your abilities is essentially a form of self harm. You're making yourself feel bad as a punishment.

I'd say the most important thing in my life is my family, but I honestly don't know if they feel the same. I know I can't quite provide them with the kind of life they deserve and they probably resent me for that.

Accepting that no one is perfect, how are you measuring quality or standard of life for your family? Is it a realistic standard? For example, it's likely unrealistic to have a standard where you were always at home, and able to focus on your families needs, but also earning enough money for them to be without wants.

Apart from that I have my grandparents and mother and all three are in rapidly declining health.

This also sucks, but there's nothing you can do here besides making good use of the time you have left. What would that be, for you, and for them?

If I was Sisyphus right now, I'd push the boulder to the top of the hill and let it roll over me on the way back.

I get that, truly, and I think is another reason you need to speak to medical professionals.