Firstly, divide those columns in Lat,Long (Data - Text to Columns), so you will have 4 columns Point A (Latitude, Longitude) and Point B (Latitude, Longitude).
Than convert Decimal Degree to metric coordinate (maybe from WGS84 to UTM) i dont know what systems do you use.
Than if you have four columns representing metric coordinate system, calculate distance directly in excel by using math formula sqrt((x2-x1)^2+(y2-y1)^2).
This would be my solution
Extension for converting into metric coordinates.
1- Format columns as numbers and insert the table into arcmap, create you first points by "right click - Display X,Y Data" chose Point A fields (Y-Latitude,X-Longitude), and chose the geographic coordinate system (regarding you comment perhaps is "GCS_Deutsches_Hauptdreiecksnetz").
2- with first step arcmap will create a temporary feature, right-click and save it somewhere (default geodatabase, no matter).
3- transform feature class into metric system, expand toolbox/data management/projections and transformation and chose project. Chose you feature (drag and drop by arcmap) and chose Projected Coordinate System (you case DHDN_3_Degree_Gauss_Zone_3). New feature will be created (projected)
4- Now open table of contents of new feature and calculate projected coordinates.
By adding two double fields (PointA_X,PointA_Y). Right click on PointA_X name and chose Calculate Geometry, in new table chose projected coordinate system and units Meters. Do same for Y coordinate.
5- Export your Table into new Excel Sheet, toolbox/conversions/table to excel. In new excel you will have 6 columns, your 4 old columns and two extended in arcmap.
6- Do same steps for PointsB fields from begin. And in the end you will have a table with 8 columns, 4 with geographic coordinates and 4 with metric coordinates. You can easily apply formula that i have mention above.
FelixIP have mentioned another away above, to create two shapefiles, merge and to convert to line, is almost the same just in this case you calculate in excel no need to create lines.
Whatever you chose, i suggest to calculate distance from projected coordinates.