CalTopo will give you an elevation when you mouse over or enter coordinates in the search box at the top of the website. This feature seems to be location dependent and works well in the USA and is spotty internationally. This solution requires a cellular connection. Consider a GNSS application that will show elevation on your phone or better still - download a free topographic map using an application like AvenzaPDF maps. Learn to read contour lines and figure out the gully depths that way.
Be warned. These CalTopo elevation values (and your GNSS elevations) are based on the WGS84 ellipsoid. These elevation values will differ from USGS topographic maps. Historical USGS topo maps will reference the NGVD29 vertical datum. Newer USGS topographic maps will reference the NAVD88 vertical datum. Depending on your location there may be considerable differences in the elevation values for your location. Elevation data and topographic maps from other countries may use other vertical datums to express elevation values.
10 meter spatial resolution USGS surface models have been freely available for decades. Outside of the USA the 30 m SRTM is freely available.
Modern surface model generation techniques rely on LiDAR data. Many USA Federal, State, Local, and Tribal governments freely distribute 3 ft spatial resolution surface models with z value errors in the centimeters. Here is the Washington State DNR LiDAR portal for your viewing pleasure. Check out the difference between the base map (SRTM) data and the LiDAR surfaces! Outside of the USA finding free surface model data is much more difficult.