What methods can high school students use to measure the radius of the earth? I suppose they could split into 2 groups and use the shadow method from different latitudes.

Maybe they could also use the stars on the horizon at two different latitudes.

Is measuring a ship below the horizon feasible ?

What other methods could be tried by high school students?
It is hard.  You should consider what you are willing to allow them to research and what you will ask them to calculate. 

First consider the classic "Shadow of a stick" method: On a certain day you measure the length of a shadow cast by a stick at two locations. The locations should be at the same longitude. By some trigonometry you can work out the number of degrees of latitude between the locations. By combining this with the distance between the locations you can calculate the radius of the Earth. 

The biggest problem is how will your students know the distance between your chosen locations?  Traditionally this would be done by surveying. The process of surveying is a lot harder than measuring the length of a shadow. If your students are allowed to google for the distance, they might (reasonably) argue that they could have just googled for the Earth's radius.

Moreover if students are actually going to use this method, they need to travel thousands of km North or South, this may not be practical, so your students can't actually measure the length of the two shadows.

A second method uses time. The time of sunrise (or sunset, or noon) is measured at two locations on the same latitude. This lets you find the angular distance between the locations and that can be used with the actual distance to find the radius of the Earth.  Again this requires you to know the distance between the two points. And there are the same difficulties in travelling.

Looking at the shadow of the Earth as it crosses the moon can be used to calculate the radius. The is described in a document "Measuring the size of the shadow of the Earth" The difficulty with this method is that it depends on knowledge of the size of the moon (and the distance of the sun, though this can be approximated as infinity) This knowledge is difficult to get directly. However, this can demonstrate that the radius is finite and fixed, ie "the Earth is round". 

In other words, to calculate the radius of the Earth, one needs additional information that can't be easily measured: Either the distance of two distant locations, or knowledge of the size of other astronomical bodies.