Do JPEG formats add significant overhead?
The picture you've posted most certainly didn't come directly from the image sensor – it was processed by the camera hardware, and software; it's pure dark grey. 

Anyway; JPEG (to be more specific: JFIF as the JPEG-carrying file format) is not zero overhead, and you can expect a couple kB for this kind of image to work.

There's a couple bytes of headers (it's really negligible), but then things get a bit more complicated:


For a monochrome image, the actual JPEG quality setting won't do anything to the file size; it just sets a quantization mask, and you're always hitting the DC bin with a value of nonzero for anything. No matter how that turns out, Run-Length Encoding for the rest of the bins in each 8x8 pixel block will reduce the rest to the same number of bytes, regardless of quantization.
What does matter is the subsampling mask for the channels. Stanley's answer is a tiny bit wrong w.r.t. to the three channels: JPEG does support 4:2:0 subsampling, meaning that you actually omit one channel every row. But that's a story for a different day. Anyways, since your image doesn't change spatially at all, that saves space
Entropy encoding (I think it's a Huffmann method) is applied to the rest. Since the picture is built from identical 8x8 blocks, these should be reduced  to very little data at all.
There's the special JPEG/JFIF feature of interleaving the info in a way that allows the lower frequency parts of the image to be transferred first (e.g. for slow connections; if you grew up in the 90s, you'll remember pictures that got sharper gradually while you hoped for your dialup modem to take less costly time). While these practically make little difference for "typical" photos, for your monochrome picture, they'll make matters worse and increase the file size by some 20 to 25%, maybe.


So, 6 kB is actually pretty OK, and you should probably look into whatever gave you that picture – it's definitely heavily smoothing the image, to a degree that the data you're presented has nothing to do with what the sensor sees. The sensor sees noise.