It is unfortunately not a big game with an involving storyline you requested, but if you have some spare time during lunch or an afternoon, I would suggest "Strange Adventures in Infinite Space" and its sequel "Weird Worlds: Return to Infinite Space".
You explore a map containing several star systems and have a time limit before you need to return to the home system.
Each system you visit might either contain an item, an encounter with another space traveling species (hostile or friendly) or an event like a star going nova.
You start with a rather weak ship which you can upgrade with items along the way like better weapons, shields, targeting computers, drives (both in system and FTL) and special hardware like self repair systems and a cloaking device.
You can also trade with the Klakar, or some of the other species like the Muktians if you can earn their friendship, trading in some of your old hardware for new ones if you manage to find their homeworlds.
Some species remain hostile to you or can be made passive but never really become your friends.
To impair your journey is a large purple nebular that reduces the speed of your FTL drive if you travel through it.
This can be negated by using special drives that are not affected, an item that makes a path in the nebula (but can bring a negative score) or the hyperdrive which works by transporting your ship to a targeted system.
Natural dangers include black holes but you can make this harmless with another special item and even use them then for instant travel.
Once you have returned to the home system an end score is made up of the various systems you visited, items you collected, encounters you made, and fights you won.
Occasionally a special event might take place such as the invasion of the Yellow Kawangi, a race of star system harvesters who mine star systems by making their host star go nova.
If the Yellow Kawangi should appear you must stop them before they reach the home system, but they are very well armed and you must find special equipment or items to beat them like the Limited Vacuum Collapser or the Chronium Gong which summons the five furies.
Another encounter starts when you find the Watch of Primordus, you must then seek out a mythical starship somewhere hidden on the map and use that to fly into a targeted black hole where Primordus' space station is hidden.
Your task is to kill Primordus before he re emerges in normal space and starts a campaign of conquest.
Last, a new threat is introduced with the latest patch in which you discover an alien insectiod hive which reveals the existence of a race of space traveling insects that travel from system to system to devour all living matter.
Two fleets of their appear on the screen once you have found the hive.
You can find more about this game on Schrapnel Games' website;
http://www.shrapnelgames.com/Digital_Eel/WW/WW_page.html
I hope this is what you are looking for.
