It's very possible! I'm starting on my first diy gas laser very soon. Give SAMs FAQ a good read before you jump in. I would start by building a CO2 laser because the gas has a ton of gain and just WANTS to lase. they are very forgiving when it comes to gas pressure which is very hard to get just perfect on other lasers.
If you plan on building a gas laser, do it for the fun and experience of the project. If you're looking to build a better quality or cheaper laser than one you can buy, you'll be disappointed! At least when it comes to HeNe lasers. A sealed hene built by a laser company will last thousands of hours. Most diy HeNe's are a 'flowing gas' design (or at least have a built in reusable gas fill valve, and a vacuum valve at the other end) because a diy sealed tube laser will have a pretty short life... Maybe 100 hours if you're really really lucky.