Calculating the distance between 2 straight lines in a 4d space?

I think you must be careful to define what is wanted here. The analog of this question in 3d space is to find the length of the unique common perpendicular between two skew lines. This is usually referred to as the distance between the lines. I believe that in 4d space there is the concept of skew planes -- that is, planes that are not parallel and are also non-intersecting. There should be a unique line perpendicular to both planes and its length could be called the distance between the planes.

Let's define distance between lines in 4d space before going farther on this one.

It'd be the same as you do it in 2D except with two extra variables. So say w,x,y,z are our variables. The distance would be sqrt[(w2-w1)^2+(x2-x1)^2+(y2-y1)^2+(z2-z1)^2]. I believe, if i remember correctly, the technical reason of this is because you are just finding the magnitude of a vector that is the difference of two other vectors. In other words, since vectors are moveable within the space, you can rearrange it so that the the two vectors start from the same point. After that you'd draw a third vector from the tip of one vector to the tip of the other. This vector you just have drawn is the difference of the two other vectors. You'd start with the vector who's head is being touched by the head of the vector you just have drawn, minus the other vector. The magnitude of this vector would just be the distance formula.