The order of boiling points is: ##CH_4 < CH_3OCH_3 < CH_4O < CaCO_3##
The order of strengths of intermolecular forces is: ##ion-ion > H-bonding > dipole-dipole > London dispersion##.
with stronger intermolecular forces have higher boiling points.
The strongest intermolecular force in each of the compounds is:
##CaCO_3## ion-ion attractions.
##CH_4## London dispersion forces
##CH_3OH## hydrogen bonding
##CH_3OCH_3## dipole-dipole attractions
##CaCO_3## is an ionic compound. It has the highest boiling points
Next comes methanol ##CH_4O## or ##CH_3OH##.
Methanol has strong . It will have the next highest boiling point.
Dimethyl ether ##CH_3OCH_3## is a polar molecule.
The ##C-O## bond dipoles reinforce each other so the molecule has a dipole moment.
Dipole-dipole forces are not as strong as hydrogen bonds so dimethyl ether has a lower boiling point than methanol does.
Finally the ##C-H## bonds in methane are nonpolar so the molecule is also nonpolar.
It has only weak London dispersion forces
##CH_4## has the lowest boiling point.
The order of boiling points is: ##CH_4 < CH_3OCH_3 < CH_4O < CaCO_3##
Here's a good video on ordering compounds according to their intermolecular forces and boiling points.