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I've heard that ESRI .gdb can. Are there other databases or file formats that can as well?

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This answer by @Stephen Lead indicates that you are correct and Geodatabases can store true curves. And here is the page to show you how to create them in ArcGIS.

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?

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    Right now, just my general knowledge and future reference. Not too long ago, I was delineating alluvial fans for a class project and saving them in .shps. I was kind of annoyed to have my nice curves reduced to line segments when I went back to edit. – dericke Jun 7 '13 at 3:23
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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.

http://webhelp.esri.com/arcgisserver/9.3/java/index.htm#geodatabases/using_t1450550752.htm

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With Oracle 12c the SDO_GEOMETRY provides full support for generic curves (NURBs) which includes support for Bezier curves and cubic splines.

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