I think another post from quite a while ago hit the nail right in the head when saying that you can't define "close friends" in the same way for everybody - especially when we're talking about an introvert like Luke who's not the type to get together with people frequently. That said, I do think he socialized with Lorelai plenty for his standards - they saw and talked to each other at the diner everyday and were almost always up to what was going on with the other.
Luke came over for dinner at her house and for the Casablanca movie night at Lorelai's invitation. They had lunch in the gazebo for the town basket event. They ate together in Thanksgiving every year. He went to Rory's birthday party and Lorelai's engagement party (even though I'm sure that he wasn't fond of crowded environments and he would've had declined to attend had it not been for them). Lorelai helped him paint the diner, accompanied him to Louie's funeral, helped him look for an apartment for him & Jess, bailed him out of jail, and gave him advice on Rachel, Nicole and Jess. Luke taught her how to fish, built a chuppah for her wedding, practically dropped everything to be there for her during a family emergency, shoveled her drive when it snowed, fixed the catwalk at the inn solely for Lorelai's sake and went to the Bracebridge dinner to support her. Not only does he insist on doing all the repairs in Lorelai's house (going as far as lecturing her for wasting money by hiring someone to do something he could gladly do for free) but he loans her a major amount of money without as much as a moment's hesitation.
And while it's true that they rarely hung out in a lot of formally scheduled plans, there's no doubt in my mind that if it came down to it - as in if for some reason Lorelai's circumstances changed or she couldn't come into the diner anymore, they would've worked out a way to spend time with each other.
Furthermore, I think the way they act around each other is pretty telling in its own right. They're never more comfortable and truer to themselves than when they are with each other. Lorelai confides on him in a way she can't with Rory, Sookie or Chris. And the same goes for Luke. When he's feeling troubled, when he needs a second opinion, when he just needs someone to vent to - who does he call? Who does he go to? When all is said and done, who is the one person he cares about the most? Lorelai.
In my opinion, LL are extremely close friends who consider themselves as such (hence why Lorelai is incensed when he doesn't tell her moved and Luke is so taken aback when Lorelai tells him that they are not friends). That's actually the reason I ship them - they love each other both romantically and platonically, but the platonic part was just as important. As a matter of fact, their closeness lent credence to their reluctance to acknowledge their feelings and pursue a romantic relationship - they were practically family to each other. Why on earth would they want to jeopardize that?