Zenith 1M30T20 (12" B&W from 1966)

[old_tv_nut]

Barkhausen oscillation is a VHF oscillation that occurs in tube horizontal outputs. A line in a solid state set is also caused by something in the horizontal sweep, but it may be something entering via a power supply or ground path to the video or by RF.
If the line looks ragged, it's RF, if smooth, probably getting in through the video section. In any case, you can verify that it originates in the sweep section by turning the horizontal hold and seeing that the picture moves left and right a bit but the line stays in the same place.

If there are flying wires that can be dressed differently, you can try moving them and see if that affects the line.