The other day I was asked - Do Chromebooks have a packet capture function?

In Developer Mode you can enter the BASH shell (Ctrl+Alt+T > shell) and type "packet_capture" to start capturing outgoing packets (reference http://www.howtogeek.com/170648/10-commands-included-in-chrome-oss-hidden-crosh-shell/).

But those who don't want to have their device in Developer Mode (as is the case with most Enterprise or Education situations), there are two other methods.

If you're only wanting the traffic from the page you're loading, you can use the Developer Tools (Ctrl+Shift+i) and goto the "Network" tab. From there, you can record the page events, where the outbound requests are going and what (if anything is being blocked). Checking the box to "Preserve log" helps when navigating to multiple pages or reloading. You can right click in the table under Name and save the collection (including loaded resource) as a HAR or HTML Archive file.

Perhaps you're more suspicious of something installed on your Chromebook (I'm thinking a suspicious extension - or even traffic from within your Crouton chroot.). chrome://net-internals can capture all traffic that goes out any network port and gives the chance to save and examine the resulting file. If you goto the "Capture" dropdown and check the "Include the actual bytes sent/received." box, the file includes actual outgoing and incoming packets, though the file will be very large.
Shared publiclyView activity