Your data should be OK and recoverable - but you might need a toolkit!
If the machine boots up (but you just can't see it), it may be possible to view the contents by connecting it to another machine on a network. However, this will depend whether the machine and/or drive is set up to allow that, which wouldn't be the easiest thing to change if you can't see what you're doing.
Failing that, the electronic connections to a hard-drive are pretty standardised, so you could remove the hard-drive (HDD) from the broken machine and either install it into another machine, or probably easier, into an external HDD enclosure (effectively converting it into an external USB drive that you could use on any machine.) Either would be routine jobs for any repair shop. If you don't intend to have the old machine repaired, the hard-drive enclosure route is a pretty easy DIY job, as you don't have to worry about breaking the old machine, and HDD enclosures are much simpler than the inside of a whole computer (posting a quick pic of the hard-drive's connections would enable us to tell you what kind of enclosure you'd need - they're usually cheaper than what a repair shop would charge you for data recovery.)
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