I've heard that ESRI .gdb can. Are there other databases or file formats that can as well?
While not a GIS format, I know Adobe Illustrator can handle Bezier curves. Also not a GIS format, but getting close, is Microstation DGN files.
It looks like PostGIS can store curves. But I'm not sure if these are Bézier curves.
If you have a look in the documentation for FME's Readers and Writers you might be able to find more. I did a search for "Bézier" and it only came up with the Microstation format. However, if you search for "Curve", you get more results. However, many of the results state that:
"A curve segment may be a circular arc segment or a linear line segment."
Can I ask what you're using Bézier curves for?
Oracle and ArcSDE can
When storing geometry in the database, ArcSDE populates the SDO_GEOMETRY value from an ArcSDE API object called SE_SHAPE. The SE_SHAPE object can contain simple and complex geometry that may include elevations, measures,* CAD data, annotation, and surface patches. The SDO_GEOMETRY data type supports a subset of these geometric properties. Because there is not a one-to-one mapping of the components in the SDO_GEOMETRY and the SE_SHAPE object, ArcSDE follows a set of rules when storing ArcSDE data in Oracle Spatial tables.
*Converts noncircular arcs (cubic spline, Bezier) to straight-edged linestrings. When the feature class is registered with the CAD entity mask (c), also store the curved representation in SE_ANNO_CAD_DATA.
With Oracle 12c the SDO_GEOMETRY provides full support for generic curves (NURBs) which includes support for Bezier curves and cubic splines.