 Compact Kakeya construction from small projections

Choose a compact $K\subset [0,1]\times \mathbb{R}$ with full horizontal projection and zero-length projections $y+tx$ for $t\in [0,1]$. Then the displayed set is compact, has zero area by the Fubini theorem, and contains a segment of every slope in $[0,1]$. Such parameter sets exist by the Baire category theorem in the Hausdorff metric: fine parallel segments approximate any parameter set while making a fixed band of projections uniformly small. Finite grids and a countable intersection make every required projection null. Four rotated copies supply every planar direction.