So I made a rookie mistake. I created a point feature class in ArcMap 10.6 without assigning a coordinate system. I put in a bunch of points and then went to export it and realized my mistake. I tried defining the projection but its projection is unknown so that doesn't do anything and I tried project(data management) but I get 000517 The coordinate system is not defined for the input dataset. Is there any way to salvage my mistake?

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Yes. These data will be referencing the same coordinate reference system (CRS) of the data frame at the time of creation. Use Windows Explorer and go to the directory that stores your shape file. Delete the .prj file. Open ArcGIS and identify the coordinate system of the data frame. Then use the Define Projection tool to create a new (correct) prj file in the correct CRS.

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