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How do I calculate perimeter of a large vector polygon (polygon set) which spans almost half the globe? I guess regular calculations do not produce accurate results in this case.

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In QGIS, it seems that if you have the project coordinate reference system set to an ellipsoid and not a projection, then calculating the perimeter via the field calculator will automatically use the ellipsoidal value:

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In ArcGIS, the tool Add Geometry Attributes appears to give you an entire suite of geometrical and geographical calculations.

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You can achieve that using web mercator projection and addGeometryAtributes form ArcGIS

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