Power issues at Bowdoin mean the Blue Line is like snowed in

There is still a stub track and tunnel space that extends beyond the Bowdoin loop toward Charles Station.

The original configuration had that stub track exit onto Cambridge Street. The closed and capped portal is in the vicinity of the fire station or just inbound of it. The track then went up an incline and connected to the Red Line at Charles Station just inbound of the current station platform where the present incline is. The reason for this was the original trains on the Blue line, then essentially streetcars, were towed onto the Red line and repaired at the car shops just off Harvard Sq, now the location of a hotel. When the Blue line eventually went to high-platform trains and the shops at Orient Heights were built, the connector was no longer needed and was disconnected and capped.

I doubt a tunnel would be possible. That area has filled land and the approach to the Charles River would both suggest a high water table. The Boston Ground Water Trust would have to weigh in were this to happen of course. And the likelihood of an elevated structure is also likely a deal killer.

From an engineering view anything is possible. Getting past the hurdles of existing laws, costs, etc is another thing.

Here's the official project map from the MassDOT website:

Note that the extension runs straight down Cambridge Street, not under MGH or any other buildings. The map also shows the existing Blue Line tail tracks, which already tunnel as far as Joy Street.

This detailed project description shows the construction outline at Charles Circle. The staging area is for a tunnel boring machine that will travel under Cambridge Street to reach the existing Blue Line tunnel. It will also bore another short tunnel on the other side of the station for train storage. Note there's a crossover right before the new Blue Line "center-track" platform. So, like at Alewife, passengers simply take whichever train is leaving first.

Please, read the linked report for all the details; the Ⓣ spent a lot of money for it to be completely planned out, but they never followed through with actually building it. Besides being promised as part of the Big Dig mitigation, it makes total sense to complete this very short, missing link in the transit system, especially when it would alleviate the horrible traffic conditions around MGH and Charles Circle, and also reduce some of the congestion on the Red Line.

Perhaps one of the reasons the Ⓣ fails to pursue this project is the fact that it would be completely underground, thus no opportunity to build an "architectural statement" station like they're doing at Scollay/Government Center for the motoring public to admire.