Generally speaking, the exterior doors to your house are (or, at least, can be made) reasonably air tight. You'll want to use similar type seals for your door.
I'm thinking a product similar to this self adhesive gasket:
Image of M-D 20-ft x 1/2-in Black Door Seal Silicone Door Weatherstrip supplied by Lowes.com. No endorsement of product or vendor intended or implied. Shown as an example only.
If your door runs inside a channel, put this gasket on both sides of the channel, all the way around the door. If it runs across the surface, put it on the frame and let the door slide across it, compressing slightly as it does, for a good seal.
As noted by Graphus in his comment on your question, getting the door to slide nicely against the seals could be difficult. Also, they're really not designed for sliding, but for compression only, so the seals may wear out far more quickly than they would on a front door.
For added ease of sliding the door and longevity of the seals, you might try designing a "plug" door like you would see on a workman's van. This would slide a centimeter or so from the wall, then push in toward the wall as it closed the last cm or so, compressing against the gasket, making a seal. Making this kind of sealing mechanism would take a bit of engineering on your part and some precision routing of channels at the top and bottom for it to run in so that it can execute the "plug" motion smoothly at the top & bottom, left & right sides, at exactly the same time and by the same depth so that it provides a good seal.