Why are holding patterns in an oval shape?
I don't have any definitive information on why the oval shape was chosen. However, it seems to me to be the most practical shape given the navigational equipment available when holding patterns first became necessary. VOR, DME, and GPS technology did not exist. Holding patterns were flown, often by single pilots hand flying, with reference to an ADF needle, maybe with a little marker beacon help.

I don't know if a square-like pattern was considered, but if it was I think it would have been rejected for at least the following;


a rectangular pattern would involve 4 turns, better to keep it simple with only 2 turns.
flying any kind of a leg perpendicular to the holding radial would mean getting farther away from the radial than necessary with no advantage to be gained, and it might involve more terrain clearance problems.


Flying a circular pattern makes it more difficult to keep track of where you are in your head (no glass cockpits, indeed no HSIs even, not even rotating ADF cards at first), than an oval. Worse, though, is what could occur in some wind conditions. For example, let's say you're supposed to hold north of a fix on the 360 radial (inbound course 180). Further, lets say there's a significant north wind. You're approaching the fix on a course of 180. You cross the fix and then fly a circle with respect to the air you're in. In a no-wind condition, you'll come back to the fix. However, with a north wind, you'll come back to the course south of the fix, and will have thus violated where you were supposed to hold.

With today's navigational technology, we can of course easily fly any kind of shape we want, but the disadvantages previously mentioned would still apply.