Yes, you may be able to renew your J-1 visa at a US consulate in a country other than your home country, as a 'third country national (TCN).'
Not all US consulates accept TCN, so you would need to check in advance with the specific consulate for its TCN policy. The website of the American Embassy in that country would have visa instructions in the non-immigrant section.
There are risks; should there be delays, you would have to stay in that country until the processing is complete; you could not reenter the US, of course, until the renewal is approved.
As a point of reference, you're able to do this since you are renewing the same type of visa which you received in your home country (i.e., that of your citizenship or residence).