Is it possible to save points and multiline strings within the same shapefile? If so, how?
No, Shapefiles contains a single geometry type. (see the Shapefile specs, p4)
As said in other answer you can't have different geometry in the same shapefile.
Some format will allow you to mix geometry but it is considered bad practice (some software wont open file with mixed geometry, geotraitement expect a single geometrie type per entry...)
If you want a single file to hold different vector type have a look at geopackage or sqlite (answer to this question for more info on these two format)