The cards you put on the battlefield enter the battlefield at the same time (the action is identified using one action verb, "put"; under C.R. 608.2c, in general, you "apply the rules of English to the text" of a spell or ability when carrying out its instructions); however, in case it matters (which it doesn't in this scenario), the active player determines their relative timestamp order for the purposes of continuous effects (C.R. 613.6j). Because no value for X was chosen for Walking Ballista, Walking Ballista will enter the battlefield with no +1/+1 counters on it, so normally as a 0/0 (C.R. 107.3e, 107.2), but Chief of the Foundry's ability will give it +1/+1 (C.R. 613.4b-c, 113.6), thus making it 1/1 in this scenario from the first moment it's on the battlefield (C.R. 611.3c).
EDIT (May 12, 2019): Correctness edit.
EDIT (Jun. 6, 2020): Edited in view of rule update with Ikoria: Lair of Behemoths.
EDIT (Mar. 1, 2022): Some rules were renumbered in the meantime.
Both creatures enter at the same time. And since static abilities (like the one of the Chief) are active from the very moment the object it's on is in the right zone (usually the battlefield), and state based actions are checked only after CC has fully resolved, by the time the check happens, the Ballista is a 1/1 and survives.
On the matter of time stamps, if two or more permanents enter the battlefield at the same time, the active player chooses the time stamp order. But that is not relevant in this scenario.
; however, in case it matters (which it doesn't in this scenario), the active player determines their relative timestamp order for the purposes of continuous effects (C.R. 613.6j). Because no value for X was chosen for Walking Ballista, Walking Ballista will enter the battlefield with no +1/+1 counters on it, so normally as a 0/0 (C.R. 107.3e, 107.2), but Chief of the Foundry's ability will give it +1/+1 (C.R. 613.4b-c, 113.6), thus making it 1/1 in this scenario from the first moment it's on the battlefield (C.R. 611.3c).EDIT (May 12, 2019): Correctness edit.
EDIT (Jun. 6, 2020): Edited in view of rule update with Ikoria: Lair of Behemoths.
EDIT (Mar. 1, 2022): Some rules were renumbered in the meantime.
On the matter of time stamps, if two or more permanents enter the battlefield at the same time, the active player chooses the time stamp order. But that is not relevant in this scenario.
Former Rules Advisor
"Everything's better with pirates." - Lodge
(The Gamers: Dorkness Rising)
"Any sufficiently analyzed magic is indistinguishable from science."
(Girl Genius - Fairy Tale Theater Break - Cinderella, end of volume 8)