You ‘cup’ is probably not a ‘manifold solid’.
If you temporarily make its geometry into a group Entity Info will say ‘Solid’ in the top bar if it is.
3d-printers want solids.
A ‘solid’ is an object containing only edges and faces and every edge must have exactly TWO faces - no fewer and no more.
So, no nested groups or components, no faceless edges, no edges with one face only [e.g. around a hole or forming a ‘shelf’], no internal partition faces where some edges have three or more faces, no otherwise solid parts meeting on a common edge where that edge then has four faces, no overlaid facets [more rare but possible]…
Part of your handle has a face oriented backwards - you should never see an ‘blue’ back-sides’ !I suspect that parts of you cup have internal partitions - e.g. where the handle joins the cylinder, of where its central seam has a common internal face ?
Edit the group.
View with Hidden Geometry ON., and/or use Xray mode and/or temporary Section-Planes to access the internal parts and manually delete problematical facets etc.
Thomthom made a Solid-Inspector plugin which highlights such ‘un-solid’ problems.
I have SolidSolver which tries to fix them - although it is quite possible to make an insoluble object! - so manual fixing is needed.
Once you have a ‘solid’ group you can explode it in the knowledge that the exported geometry should be compatible with 3d-printing.
Incidentally - OBJ is one format, but STL format is more standard - there is an Exporter freely available in the Extension Warehouse…