I ran a few tests and can reproduce the problem at the service directory so perhaps this is an issue with the service? Might be best to submit an issue to support so they can work through this with you and troubleshoot your service setup.
Here's the test I used to repro
1. Navigate to the service directory
Execute Task (Export Web Map)
2. Enter the following in the Web Map as JSON field
{"mapOptions":{"showAttribution":true,"extent":{"xmin":-13074899.231001616,"ymin":4015941.6611683434,"xmax":-13073257.02922219,"ymax":4016397.894680882,"spatialReference":{"wkid":102100,"latestWkid":3857}},"spatialReference":{"wkid":102100,"latestWkid":3857},"scale":4513.988704998488},"operationalLayers":[{"id":"layer0","title":"layer0","opacity":1,"minScale":591657527.591555,"maxScale":1128.497176,"url":"http://services.arcgisonline.com/ArcGIS/rest/services/World_Imagery/MapServer"},{"id":"layer1","title":"layer1","opacity":1,"minScale":591657527.591555,"maxScale":1128.497176,"url":"http://services.arcgisonline.com/ArcGIS/rest/services/Reference/World_Boundaries_and_Places/MapServe..."},{"id":"graphicsLayer2","title":"graphicsLayer2","opacity":1,"minScale":0,"maxScale":0,"layerDefinition":{"drawingInfo":{"renderer":{"type":"simple","label":"","description":"","symbol":{"color":[255,255,0,255],"outline":{"color":[255,255,0,255],"width":2,"type":"esriSLS","style":"esriSLSSolid"},"type":"esriSFS","style":"esriSFSBackwardDiagonal"}}}},"url":"http://sampleserver6.arcgisonline.com/arcgis/rest/services/PoolPermits/MapServer/1"},{"id":"map_graphics","opacity":1,"minScale":0,"maxScale":0,"featureCollection":{"layers":[]}}],"exportOptions":{"outputSize":[800,1100],"dpi":96},"layoutOptions":{"titleText":"Pool Permits","authorText":"Made by: Esri's JS API Team","copyrightText":"<copyright info here>","scaleBarOptions":{"metricUnit":"esriKilometers","metricLabel":"km","nonMetricUnit":"esriMiles","nonMetricLabel":"mi"},"legendOptions":{"operationalLayers":[]}}}
3. Enter a valid format (PDF, PNG32, PNG8, JPG, GIF, EPS, SVG, SVGZ)
4. Enter MAP_ONLY in the layout template then hit 'Execute Task (Post)' and not that it works. Try again after changing the layout template to Mono8X11Land_Info and note that it times out eventually.