I am serving some large MPEG4 files internally, via Apache http, and I am finding that the Chrome browser will try redownloading the whole video every time I scrub through the timeline. Is there any software that will deal with serving video that doesn't result in the whole download being initiated again?
You can use ffserver, which is installed with FFmpeg.
ffserver is a streaming server for both audio and video. It supports several live feeds, streaming from files and time shifting on live feeds. You can seek to positions in the past on each live feed, provided you specify a big enough feed storage.
If you are serving very large files that take long enough to download where a connection risks being severed, then use a synchronization tool like Dropbox, Google Drive, Amazon Drive for teams. That can cost money to set up, OR you can store them in an Amazon Web Services S3 bucket and have users download them with the AWS Command Line Tools. Those tools handle shaky connections very well.