Old school -
I haven't done Recombinant work in years (+ decade), so I don't know how they do it now.
In the past:
small pieces of DNA can be separated using polyacrylamide gels( 10 - 500 bases) - single strand
due to the small size of the pores, these can't separate large pieces of DNA
agarose gels then come into play - up to certain point (300 to 10 maybe 30000 bases), extremely large pieces of DNA ( if memory serves me right 30000 to 2 million bases) needs a process called pulsed-field gel electrophoresis
Since DNA is negative and all of these gels have pores, etc - the DNA is separated by size (bases). To determine the size, a standard is used. One buys them - some DNA cut by a restriction enzyme like BamHI, EcoRI or TaqI.
We would know the sizes of the DNA pieces in the known. We create a log plot off that data based on size (# of bases).
then we could estimate the sizes of our sample DNA.
Problem - potentially 2 pieces of DNA could be the same size. In protein land, we used to use a 2 dimensional gel, I wonder if they do that now to eliminate this problem.
As for using weight, size does relate to weight - but purines and pyrimidines do have different weights - on a large piece of DNA that weight may not make much of a difference, but on a very small piece of DNA ( a couple of nucleotides) it does.
I apologize if this post is information you already know, I don't know your background on this material.