First of all, note that
with the usual definitions, it is in fact not possible to construct a polygon with only right angles and an odd number of corners. One way to see this: imagine walking around the polygon and keeping track of whether you have turned an odd or an even number of quarter-turns overall; when you get back to where you started you are facing the same direction as before and that number must therefore be even.
clearly we are looking for some unorthodox definition of something. The only really plausible candidate, I think, is "polygon". For instance, we might consider a polygon lying on some curved surface. Then our polygon would no longer need to be planar, and the argument above wouldn't work.
Here are two versions of this. First,
allow the edges of the polygon to be geodesics on whatever surface we have. In that case, there's a particularly simple solution: let the surface be a sphere, on which the geodesics are great circles, and then consider a spherical triangle with three right angles. (One way to make one: take three planes through the centre of the sphere all intersecting at right angles; these divide the sphere into 8 portions each of which is a spherical triangle with three right angles.)
you might find that unsatisfactory because those great circles aren't "really" straight lines, they're just "locally straight" within our curved surface. So, as an alternative, let's see whether we can construct a surface with enough actually straight lines on it to make this work. This is actually pretty easy. Let's construct the "polygon" first. In ordinary 3-dimensional euclidean space we'll start at (0,0,0) and then go to (1,0,0), then (1,1,0), then (0,1,1), then (0,0,1), then (0,0,0) again. Five straight-line sides, right angles between consecutive ones. Now for our surface. Start at the corners. Place a little square patch at each corner, in the plane spanned by the two edges meeting (at right angles) there. Then join adjacent patches with strips running along the edges, twisting a little as required. If you sketch this you will see that the result is a nice nonplanar pentagonal strip with a pentagonal "hole" inside it, and no weird nonorientable Moebius stuff going on. We can fill it in and extend it out pretty much however we like. The result is a surface on which those five straight line segments live happily, forming a right-angled pentagon.