alpha = make active bets.
you decide a stock is undervalued - buy it. if it goes up - you gain more.
you decide a stock is overvalued - sell it. If it goes down further - you gain more.
When you make a bet on two companies in the same industry - go long on one, go short on another, with the same $ amount - you end up with a market neutral long short strategy - and if your bets work right - Long goes up, short goes down - you get two alphas from that decision.[Of course if long goes down, short goes up - you lose, but that normally should not happen if your active bets were right].This strategy provides you with zero beta exposure - since the market is the same. You have unsystematic exposure.
You are right that the beta exposure is a Long only investment. Issue with a long only investment - especially with regards to “indexed” investment - is that your best bet for a stock you like a lot = 100% long that stock. For a stock you do not like - you put 0% investment (an active bet of -X% where X% is the weight in the index). That is all you can do. So you are limited.
Now - say you wanted both alpha and beta from the same market … you start to look at combining the two strategies above.
Given that you could also do get alpha from one market and beta from another - the alpha part becomes PORTABLE as well.