MP4 is a container. It may even be able to contain MPEG-2 video, I guess; but that's not what you want.
You may want to convert to MPEG-4/AVC video (a.k.a. H.264). This supports interlacing, so one option is to retain this attribute. Playback on a modern flat screen TV set should be supported with good motion adaptive deinterlacing during display.
For playback on a PC, different converters support a variety of deinterlacers. Handbrake may not be one of the most versatile, though. And before we recommend you one, you really need to check your source whether it is really interlaced, or instead telecined: The latter can be converted back (almost losslessly) to an originally progressive content. That method is not at all related to "deinterlacing". But it also works only if your material is a result of a Telecine process. You need to test that.