They will remain, but have no defined physical form
Elementals aren't inherently the product of magic: they exist before a spell is cast, and presumably will exist after its effects fade. However, the embodied form of an element we think of is different from the natural form of elementals. Note the following statement on elementals on page 123 of the Monster Manual (bold added)
On its home plane, an elemental is a bodiless life force. Its dim consciousness manifests as a physical shape only when focused by the power of magic.
Although the existence of elementals is not inherently magical, the familiar form they take on that turns them into creatures whose stat blocks are listed in various books are exclusively the product of some magic or another.
On its own native plane, an elemental isn't a creature summoned by magic. However, the rules on antimagic field state (bold added):
A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.
The spirit of the elemental is not created by magic, but the physical form that makes it a defined creature (e.g. an Earth Elemental, an Invisible Stalker, etc.) is created by magic. Thus, that familiar creature will cease to exist in its current form while it is within the antimagic field. Once the field has moved away or ended, the elemental's disembodied spirit will once again form into a physical creature.