Cut street for cars and add pedestrian barrier

As you can see from this routing example cars are already forbidden to take that path. However, you could add the tag motor_vehicle=no or motorcar=no to the path to be sure every routing engine won't make cars pass on that path.

You can add a node with the tag barrier=fence. I'm not sure if you can add foot=yes bicycle=yes tags so that only pedestrians and bikes can pass through. However, according to the StreetView image you provided, I can see on the left of the fence that there's a little gate like that in my beautiful painting below. If there's a gate, you could add a node with barrier=fence foot=yes bicycle=yes tags.

You can add the barrier to the road, and potentially convert a small section of that road to pedestrian for the routers that don't recognise barriers.

A list of barrier features is available here. It needs to be placed so that it is on a node that is part of the way that is blocked but not on ways that aren't. I.e. Not on the intersection unless all routes through the intersection are blocked.

The road segment could be highway=path or =footway.