Draw the boundary on the map and read the area in every Nepali and metric unit. Live as you move each point.
The area is calculated geodesically — on the curve of the earth, not by multiplying latitude by longitude, which at Nepal's latitude overstates area by about 13%.
What it cannot do is tell you where your boundary legally runs. You are tracing what you can see on imagery that may be a year old and a metre or two out of alignment. Treat the result as a check on a figure you already have, not as a replacement for one.
Your polygon, its area and anything you save stay in this browser. The map tiles and the location search do go to their providers — OpenStreetMap for the map layer, Esri for satellite imagery, Nominatim for search — because that is how any map works. Those requests tell them which area you are looking at. Everything else stays here.