Inclusion Criteria:

* For Part 1, Part 2, Part 3: Metastatic castration resistant prostate cancer (mCRPC) with histologic confirmation of adenocarcinoma (adenocarcinoma with small-cell or neuroendocrine features is allowed) with prior exposure to at least one androgen receptor (AR) targeted therapy (for example [e.g.], abiraterone acetate, enzalutamide, apalutamide, darolutamide). In addition: Part 1: prior taxane or other chemotherapy is acceptable but not required. Part 2a: prior taxane or other chemotherapy required, Part 2b: no prior taxane or other chemotherapy, Part 2c: mCRPC that has progressed after prior treatment with lutetium Lu-177 vipivotide tetraxetan, with or without prior chemotherapy, Part 3: prior taxane or other chemotherapy is acceptable but not required & For Part 4a: metastatic HSPC, For Part 4b: disease that can be treated with less than or equal to (<=) 5 radiation fields and no visceral metastases
* Parts 1, 2 & 3: Prior orchiectomy or medical castration, or, for participants who have not undergone orchiectomy, must be receiving ongoing androgen deprivation therapy with a gonadotropin releasing hormone (GnRH) analog (agonist or antagonist) prior to the first dose of study drug and must continue this therapy throughout the treatment phase. This criterion does not apply to Part 4
* Palliative radiotherapy (e.g. soft tissue lesions) must be completed greater than (>) 2 weeks prior to start of study drug except for palliative radiotherapy for pain (e.g., bone pain), which may be used any time prior to first dose
* Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* Adequate organ functions as reflected in laboratory parameters

Exclusion Criteria:

* Prior treatment with radium Xofigo (Ra 223 dichloride), strontium, samarium, or other radioconjugate therapy, other systemic anti-neoplastic therapy <=30 days prior to the first dose of study drug except for luteinizing hormone-releasing hormone agonists/antagonists or GnRH agonists/antagonists. Novel androgen axis drugs <=14 days prior to the first dose of study drug. In addition: Part 2b: Must not have received prior treatment with chemotherapy (eg, docetaxel) or poly ADP ribose polymerase (PARP) inhibitors, Part 2c: Prior treatment with lutetium Lu-177 vipivotide tetraxetan is required, but must have been completed >42 days prior to first dose of study drug, Part 3: Must not have received prior treatment with JNJ-78278343, Part 4: Must not have received ADT or AR-targeted therapy less than or equal to (<=) 56 days prior to first dose of study drug
* Known history of myelodysplastic syndrome, leukemia, or hematological malignancy with features suggestive of myelodysplastic syndrome/acute myeloid leukemia at any timepoint
* Toxicity from prior anticancer therapy has not resolved to baseline levels or to Grade <= 1 (except alopecia, radiation tissue fibrosis, or peripheral neuropathy)
* Known allergies, hypersensitivity, or intolerance to JNJ-69086420 or its excipients and protein therapeutics. For Part 3, known allergies, hypersensitivity, or intolerance to JNJ-78278343 or its excipients or protein therapeutics
* Active or chronic hepatitis B or hepatitis C infection