Can I build a hot tub using lumber?
You need to calculate the outward force.

Assume 8' x 8' x 3' hot tub, filled to the top.

3 ft of head * 0.424 * 1(specific gravity of water) = 1.302 psi

That's the pressure at the bottom.  Obviously the pressure at the top is 0.
If I were to sketch the pressure gradient, it would be a triangle, so we can easily calculate the average pressure as 1.302 /2 = 0.651 psi.

That pressure is acting over an area of 8x3x 144 = 3456 si.

So the outward force on the side wall is 0.651 lb/si * 3456 si = 2250 lbs.

For tipping purposes, take this force to be acting as a point load at a height of 1 ft from the ground (Centroid of the pressure gradient triangle).

So, whatever you build will need to support an outward push of 1 ton on each wall.  You might be able to get away with it using some sort of steel banding, but I wouldn't consider it with just lumber -- your corners are going to rip apart.

Look into buying a pre-formed fiberglass hot tub liner.  That will support the outward forces, and all you have to do is build the appropriate frame to support it.