It depends a little on what you mean by 'Live-Stream'.
A number of devices exist that will send your location at regular intervals, which can then be tracked via a website. Probably the most common of these are the Garmin InReach range of devices, and the Spot trackers.
These devices generally also allow you to send and receive SMS messages, either directly via the device itself, or via a connected Smartphone (not using the smartphone coverage). Prices are generally fixed per month, and generally very affordable (eg, $10-40 per month for Spot depending on the service)
If you are thinking of actually streaming audio or video, the only real options are satellite phones, which generally run on one of three satellite systems - Thuraya, Iridium, and Globalstar. Prices for such services vary depending on the provider and the package, but generally start at "insane" and go up from there. Voice calls are generally around $1+ per minute (a little cheaper if buying in bulk), and data rates vary dramatically but start at several dollars per megabyte. Unless you are, for example, a new organization like CNN, these options are almost certainly not viable. The devices themselves are also expensive, especially those that support data, starting at around $1,000 and going up from there.
Cheaper satellite internet is available - such as the upcoming Starlink systems you mentioned, however these require (relatively) large base stations and/or antennas/dishes which would not be feasible for a hiking trip.