Elite:Dangerous is the newest installment of the same series as the C64 game, and is a sandboxy-sort of online multiplayer thing.
Star Citizen is another (in dev; currently you can buy into the public alpha-sort-of thing) one, produced by Chris Roberts of Wing Commander fame. It's being sold as a "first-person universe", because you can also leave the cockpit of the space ship and run around places, optionally shooting things in an FPS sort of manner.
Of the two E:D can be played on slightly lower-end PCs, but SC is pretty demanding of a computer, especially in its current mostly-unoptimised alpha state.
Both of the games are pretty fun, but have their problems too.
E:D can be a lot of fun in small doses once you get your feet wet, but can be pretty grindy to start out with and gets kind of shallow and repetitive later. Also it's super-amazing to play in VR, if your computer can handle it.
SC is looking hugely promising and the current alpha gameplay is a lot of fun if you can deal with the bugs, but has a lot internet drama surrounding it. Said drama is mostly related to the game taking forever to develop, and the fact that you can currently pay (sometimes rather insane sums of) real money for getting your hands on the space ships currently available. The devs are insisting the buy-ships-for-money thing is only happening during development to help fund the process and will no longer be possible after proper launch, but we'll see how that works out in practise.