When I originally moved to the UK in 1999, our company house was located in Eastleigh. I didn't particularly like the area. It felt, at the time, like a poor suburb of Southampton with little charm. It didn't have the greatest reputation with the British folks in our office either. (Beastleigh) Of course, that was a decade ago now, so you probably should see if you can get a more recent opinion. Our office was very close by, in Otterbourne, and another lovely village of Chandler's Ford was also close and quaint.
That said, Eastleigh had all of the things you'd need to have a convenient life - Groceries close by, a main street with pubs and restaurants (and kindof icky nightclubs), a great and new (at the time) sportscenter with a big pool and some nice sized parkland. Because the housing in Eastleigh really runs into the surrounding areas, I'm sure that you could find nicer and less nice areas to live within the Eastleigh vicinity and it's pretty close to Winchester too, which is a lovely town with a nice high street and college and beautiful cathedral.
Good Luck!
