Yes, you can. Obstinate Baloth and Big Game Hunter both have replacement effects, so the baloth will end up on the battlefield at the same time that the hunter will be in exile. Then, madness will trigger, you cast the spell, and the creature enters the battlefield. By that point, the baloth will be a legal target for the ETB trigger.
In this case, each player discards a card simultaneously, so in this case the active player makes all "choices required", such as the choice of which card to discard, then every other player does the same in turn order (C.R. 101.4). Thus, if it's your turn, you choose which card to discard before other players choose; while doing so, however, you're not required to state the identity of which card you're discarding from your hand, but even so, you "must clearly indicate which face-down card [you're] choosing" (C.R. 101.4a, 400.2).
Therefore, in general, no matter whose turn it is, no player is guaranteed to know which cards each other player will discard due to Liliana's first ability until all those cards are discarded.
If Obstinate Baloth and Big Game Hunter are discarded this way, Obstinate Baloth will be on the battlefield even before Big Game Hunter's madness triggered ability will even go on the stack (C.R. 116.3b, 116.5, 702.34a), let alone get to resolve.
3/14/2017: When Liliana's first ability resolves, first the player whose turn it is chooses a card to discard, then each other player in turn order chooses a card to discard, then those cards are discarded simultaneously. No one sees what the other players are discarding before deciding which card to discard.
It is your turn when you activate Liliana's ability, so your discard choice is made first, and then your opponent makes their discard choice. Also, the choices are made in secret (so you just set aside on the table one of the cards in your hand, still face-down, or some similar arrangement). Only then both cards are revealed and put on the graveyard at the same time (or battlefield for Obstinate Baloth and exile for Big Game Hunter, due to their abilities).
If you did luck out and selected Big Game Hunter before knowing the opponent was discarding Obstinate Baloth, then yes, the BGH will be able to kill the Baloth. Baloth enters the battlefield instead of going to the graveyard, and at the same time BGH goes to exile. Then you can cast it for its Madness cost, and when it resolves, it enters the battlefield and its ability may target the Baloth.
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Therefore, in general, no matter whose turn it is, no player is guaranteed to know which cards each other player will discard due to Liliana's first ability until all those cards are discarded.
If Obstinate Baloth and Big Game Hunter are discarded this way, Obstinate Baloth will be on the battlefield even before Big Game Hunter's madness triggered ability will even go on the stack (C.R. 116.3b, 116.5, 702.34a), let alone get to resolve.
It is your turn when you activate Liliana's ability, so your discard choice is made first, and then your opponent makes their discard choice. Also, the choices are made in secret (so you just set aside on the table one of the cards in your hand, still face-down, or some similar arrangement). Only then both cards are revealed and put on the graveyard at the same time (or battlefield for Obstinate Baloth and exile for Big Game Hunter, due to their abilities).
If you did luck out and selected Big Game Hunter before knowing the opponent was discarding Obstinate Baloth, then yes, the BGH will be able to kill the Baloth. Baloth enters the battlefield instead of going to the graveyard, and at the same time BGH goes to exile. Then you can cast it for its Madness cost, and when it resolves, it enters the battlefield and its ability may target the Baloth.