Yeah, google drive sucks in that aspect. It doesn't save local copies of most files that do not count towards your limit (mostly docs, sheets, etc) anywhere in our hard drive. Not that we can easily access anyway. Also, even its "offline" feature, while it seem to work some times, it often fails.
Oddly enough, video files (smaller than 15 min they say) also do not count to your drive quota limit. But you do have a local copy of them! Go figure...
In any case, there are plenty of tools and mostly services out there you can use to make that local copy. Plenty. And each one do it in a different peculiar way. Most of them will help syncing with other services, such as Dropbox (referral link).
This is my on going personal research listing the most interesting ones I've found:
- CloudHQ connects services. 15 days free trial. Cheapest plan U$ 99 per year.
- Otixo, connects services. 14 days free trial. Cheapest plan: U$ 39 lifetime.
- CloudKafe, simply do search. Free (catch?). Won't work for this case!
- InSync, expands google drive. 15 days free trial. Cheapest plan: U$ 15 lifetime.
- OwnCloud, virtual desktop. Open source, offers unnamed priced business.
- JoliOS, online virtual desktop. Open source. Won't work for this case! (I think)
- Zapier, connects services with tasks. Very limited free plan. Cheapest plan: U$ 165 per year.
- IFTTT, connects services with tasks. Completely free, somehow. Won't work for this case!
I'm only trying to elucidate why I would, right now, highly advise using OwnCloud. In my head, if you're personally using Google Drive, makes no sense to pay for a smaller service. And if you're with business Google Drive, makes no sense to pay more for a smaller service. All the reasons why OwnCloud makes the most sense.
But if you're not willing to go through all the trouble of the 1 time techie setting up, simply take InSync. They provide a great service for some years already and will likely stay around.
Disclaimer: I haven't personally tried them all!