Spin transport in carbon-based materials has stimulated much interest due to their ballistic conductance and a long phase coherence length. While much research has been conducted on individual carbon nanotubes, current growth and placement techniques are incompatible with large-scale fabrication. Here we report nonlocal spin injection and detection in single wall carbon nanotube networks. We observe spin transport over a distance of 1 um, and extract a spin diffusion length of 1.6 - 2.4 um with an injected spin polarization from CoFe into nanotube network of 18 - 41%. Our observations demonstrate that spin transport is possible in carbon nanotube networks due to the formation of natural tunnel barriers between nanotubes and metallic contacts.