We need a lot more detail to be able to answer with any chance of helping you!
Which version of MEP? Computer Specs? OS? CPU and RAM? HDD storage space? Type of DVD writer? What exactly is the error message? You say you have "trouble encoding the movie". Is the problem one of encoding or is it, as your post title suggests, one of the actual burn failing? What is the exact sequence of events you carry out which leads to this error message appearing?
We are also users, rarely does MAGIX support staff answer here, but will do our best to suggest solutions if you can supply more details.
As I said before we are all users so, unfortunately, the error log is not easily understandable. However, it appears to suggest that the actual DVD files are being successfully created (mention of .VOB file). Has this/these been stored in the location the log suggests? If so, another DVD burning program might be able to complete the task for you. (In my case I only ever allow MEP/VPX to create an Image File and then use a different program to burn the actual DVD).
Other than that; I assume from the log you are attempting to create a DVD Image file, rather than burn the DVD directly? If so, did you choose "Image Recorder" as the output on the burn screen?
Finally, just one other thought; you clearly have plenty of HDD space but are you storing the original video files on the same drive as your OS and apps? As a general rule the advice is to store your OS and apps on one drive and all your data (video) files on another. (I actually have one HDD for OS only, one for apps only and all data etc. stored on a variety of others. This certainly seems to give me much greater stability and far fewer problems of the type you are experiencing. Yes, I have experienced the "DVD Burn failure" several times before, but not recently).