Currently, there is no option to export entire web maps from ArcGIS Online. The export option is only available for backing up individual feature layers from ArcGIS Online to a local device. However, depending on the desired data format, there are several workarounds to back up web maps from ArcGIS Online.
Save the web map as a project package to a local device
Save the web map as a map package to a local device
Download the web map as a .pitemx file to a local device
In ArcGIS Online, download the web map as a .pitemx file to back up the web map as an ArcGIS Pro portal item on a local device.
The web map is automatically downloaded as a .pitemx file to the local device.
Use the Extract Data tool in ArcGIS Online to export map layer packages to a local device
Note: The Extract Data tool is only supported in Map Viewer Classic.
Note: The Extract Data tool packages the layers of a web map as a dataset, not the web map itself.
In ArcGIS Online, use the Extract Data tool to package the layers of a web map as a downloadable dataset to a local device. Use the following steps to pack and export layer packages using the Extract Data tool:
Export the web map as a georeferenced PDF file to a local device
In ArcGIS Online, export the web map as a georeferenced PDF file to a local device using ArcGIS Web AppBuilder.