The sets you are calling small are commonly referred to in the literature as "strong measure zero sets." The Borel conjecture is the assertion that any strong Borel set is countable.

This is independent of the usual axioms of set theory. For example, Luzin sets are strong measure zero; if MA (Martin's axiom) holds, then there are strong measure zero sets of size continuum. In fact, both CH and ${\mathfrak b}=\aleph_1$ contradict the Borel conjecture.

However, the Borel conjecture is consistent. This was first shown by Laver, in 1976; in his model the continuum has size $\aleph_2$. Later, it was observed (by Woodin, I think) that adding random reals to a model of the Borel conjecture, preserves the Borel conjecture, so the size of the continuum can be as large as wanted.

All this is described very carefully in Chapter 8 of the very nice book "Set Theory: On the Structure of the Real Line" by Tomek Bartoszynski and Haim Judah, AK Peters (1995).

The sets you are calling small are commonly referred to in the literature as "strong measure zero sets." The Borel conjecture is the assertion that any strong measure zero set is countable.

This is independent of the usual axioms of set theory. For example, Luzin sets are strong measure zero; if MA (Martin's axiom) holds, then there are strong measure zero sets of size continuum. In fact, both CH and ${\mathfrak b}=\aleph_1$ contradict the Borel conjecture.

However, the Borel conjecture is consistent. This was first shown by Laver, in 1976; in his model the continuum has size $\aleph_2$. Later, it was observed (by Woodin, I think) that adding random reals to a model of the Borel conjecture, preserves the Borel conjecture, so the size of the continuum can be as large as wanted.

All this is described very carefully in Chapter 8 of the very nice book "Set Theory: On the Structure of the Real Line" by Tomek Bartoszynski and Haim Judah, AK Peters (1995).