yes, it is underdetermined, I forgot to clarify, I append below the exact code. It seems like BLAS crushes with Float64 as well. Yes probably a problem with SVD.
julia> obs=10;
julia> vars=10_000_000;
julia> x=rand(MersenneTwister(0),Float64,obs,vars).-Float64(1/2);
julia> a=rand(MersenneTwister(1),Float64,vars).-Float64(1/2);
julia> y=x*a;
julia> @btime $x\$y
ERROR: ReadOnlyMemoryError()