My main MMOs at the moment are Guild Wars 2 and Star Trek Online.
Guild Wars 2 I find to be a good solo experience, but you already have that and if you're looking for something new...

Star Trek Online has star ships. It's ground combat is a bit lack luster compared to other MMO, but the ship combat is just something you can't find anywhere else. Only way to know if that is a draw for you is to try it out. We're six days into our Christmas event, and by playing the daily "ice ring race" quest twenty five days (event lasts at least a week into January so there is more than enough time if you start now) you can get a free T6 star ship (this Christmas it is a tactical Bree ship.) We get three free top tier star ships per if IF we are actively playing the game, one at Christmas, one at the anniversary event in February, and one at their July/June summer festival on Risa. So starting now would have long term benefits.

Now, just to play Devil's Advocate against the game for a bit... this game might be free to play, but there are some cash ceilings if you don't pay at all. Most noticeable is the energy credit cap of 10 mill, which you'll hit hard and fast once you're max level just from doing the "Tour the Galaxy" daily quest. Bridge Officer slots and Duty Officer Rooster limits can also be pretty low. All of that takes Zen to upgrade, and Zen takes either dilithium or real world cash to acquire. There is also inventory space and other T-6 ships to unlock with Zen, but they aren't as needed as the officer space and Energy Credit limit.

I'm not much of a socialite in any of my gaming, even MMO. (I like MMOs because other people make the world bright and colorful, not because I want to interact with them.) If you decide to take the plunge I can offer some slow support for your questions right here, and point you towards my fleet for live support. They are a very welcoming group of people. http://startrek.44thfleet.com/ Otherwise, biggest question you might need answered that you wouldn't otherwise think to ask is "Why are things dieing so slowly?" The answer is that game defaults all new star ships to balanced power levels, so shift all power to weapons as soon as your behind the helm. It should stay there until such a time you decide to outfit your ship with science powers; those will require auxiliary power to pack a punch.

would you say this game is accessible to folks like me who don't watch star trek and kmow very little about sci-fi?

Well since the biggest draw, as stated, is flying star ships... a lack of knowledge/interest in sci-fi might be a problem.

As for Star Trek itself, it obviously has a lot of draws to appeal to people of the Star Trek universe. The aesthetics, the plot lines, and several original voice actors voicing their original roles all create a theme park feel. (Though that last part is hit or miss since hearing Seven of Nine, Harry Kim, and Tovok are great but since it's a video game that means you also need to listen to them spout expository dialog.) Not having an interest in those characters might make it easier to not ignore some of the flaws behind the venire. Small things like characters not sitting down properly in chair during cut scenes, or a piece of special effect that was working fine one week suddenly breaks because someone moved a squirrel over to the left in the servers. Not to mention some of the story lines are a bit short where they need to be long but long where they may have been better short.

If any of this would make it less accessible to you... I don't think so, but it certainly might have less staying power without those preexisting star trek hooks in your system. I actually am a star trek fan, so I'm not much of a judge on how a non-fan would react. I'd just be warned that the flaw of "some thing may be too short" may actually be a bigger issue for you, as some small plot point that is thrown out of left field to steer the plot might not get enough screen time for a non star trek fan to appreciate it's weight, while a star trek fan would in turn be left squealing for five minutes just for some plot element of the star trek universe being used.