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My parents have, much to my wife's annoyance, found my old game gear which has been in a bag in their garage for nigh on 25 years.
It turns on, but there's no sound and the screen is like a tv out of tune.
Restorable?
definitely, they're still really popular, you can buy new replacement compatible parts including screens, and fully working ones fetch decent money.
Like a lot of old electronics, the capacitors go, symptoms include dodgy sound & screen apparently, could be an easy thing to try if you are half-decent with a soldering iron: https://www.ifixit.com/Guide/Sega+Game+Gear+Capacitor+Replacement/113655
Otherwise there's a wealth of information online on how to fix them/replace parts, depends exactly what's broken/wrong with it though. I see there are a number of FB groups for collectors, I'm sure someone on one of those could point you in the right direction.
Absolutely restorable and upgradeable.
Sounds like your capacitors have all blown, that's a very easy fix BUT whilst you're in there you could do a few mods. F'rinstance you can change the power board, make a couplde of cuts and install LIon battery packs that don't run out in an hour!
https://www.reddit.com/r/Handhelds/ will have a load of info, the capacitors are well documented and there are planty of places to send it if you're not confident yourself. Retrosix seem to get a lot of shit but they do have a lot of mods and tbh beat the hell out of trying to import stuff. (FWIW their capacitor kit isn't the best, there are other sellers who do decent caps at the right capacitance, that's one area I wouldn't skimp on for the marginal cost difference)
Be warned, it's a right rabbit hole!
Cap guides:
http://wolfsoft.de/wordpress/?p=270
https://members.optusnet.com.au/eviltim/ggrgb/ggrepair.html