[Update: My wife got a new apartment. She's moving out next week. It's killing me. Anyway, I think there is some good advice in this post so I'm leaving it up.]

Yes, they can. (Or not. My wife of nine years announced days after this post was made that she's leaving me, and the main reason is my inability to land steady, full-time work over the past two years. Don't listen to anything I have to say below because it doesn't work. I did all that and more, and it didn't save my marriage.)

I'd be lying if I said it was easy to be in a relationship wherein one or both partners becomes unemployed. It's not. It sucks, like so many other things about unemployment. Having been through a couple of long periods of joblessness and remained in the same relationship through both, I can speak with some authority on the subject and seen my marriage dissolve over it, I can speak to it.

How your relationship might endure the stress of unemployment (at least for a while):

• Think of your relationship as a business. You can add value to your household without a job -- that's why they call the class "Home Economics." Wealth has little to do with money. Find ways to add value while reducing expenses.
• Try hard every day. Your partner will know if you are slacking off while they work to pay the bills, and they will resent it. Do projects around the house. Take out the garbage. Look for jobs. Do something productive, though, that adds value to the home. Eventually, you might even stumble on something that will pay a few bills.
• Communicate. Tell all. Don't hide anything emotionally or financially. That's essential to a healthy relationship whether you're making a million bucks a year or living in a van down by the river.
• Respect yourself and your partner. One thing that tends to happen when couples face external stress (like financial stress) is that they start to snipe at each other. This does nothing to relieve the stress and infects formerly healthy relationships like a pernicious disease. Remember why you are with this person, what brought you together in the first place. Deal with the problem at hand -- together -- and don't let the blame game divert your energy from the mission which is to stay together and get a new job, in that order.
• Do stuff together. Believe it or not, there are a ton of free or very cheap activities that you can perform together as a couple or family. Walk to the park and feed the ducks. Pop some popcorn (real popcorn, not that microwave junk) and watch TV together. Go for a hike. Check out your local library. Etc., etc., etc. ad might-as-well-be infinitum.
• Remember your vows. If you're married, remember the "for richer and for poorer" clause. Nothing is permanent. When you're up, it just means something will knock you down, and when you're down, do what you need to do to pick yourself up.

That's all I can think of now. It's a good start. Come back for more. In the meantime, feel free to add your own in the comments.