You cannot embed all paracompact spaces in this way. Even if we allow at most countably many non-zero coordinates: in that case we get subspaces of $\Sigma$-products, which have been well-studied in general topology and functional analysis. Compact subspaces of these are called Corson compact spaces, and none of these map onto $[0,1]^\{\omega_1}$. So the latter space is an example of a compact Hausdorff space (so paracompact etc.) that cannot be embedded into a $\Sigma$-product of copies of $I$ (or $R$).

You cannot embed all paracompact spaces in this way. Even if we allow at most countably many non-zero coordinates: in that case we get subspaces of $\Sigma$-products, which have been well-studied in general topology and functional analysis. Compact subspaces of these are called Corson compact spaces, and none of these map onto $[0,1]^{\omega_1}$. So the latter space is an example of a compact Hausdorff space (so paracompact etc.) that cannot be embedded into a $\Sigma$-product of copies of $I$ (or $R$).